Subgroups — #28371963

Order 13952.

364 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295580	C₂	1	1	yes	maximal, characteristic
2	#20295156	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295154	C₂	1	2	yes	maximal, complemented
4	Q₈×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295150	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295150	C₂	1	2	yes	maximal, complemented
7	#20295051	C₂	1	2	yes	maximal
8	#20295048	C₂	1	1	yes	maximal, characteristic
9	#20294983	C₂	1	2	yes	maximal
10	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
11	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182012	C₂²	1	2	yes	—
13	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
15	#16812698	C₄	1	1	yes	characteristic
16	#16812697	C₄	1	1	yes	characteristic
17	#16812689	C₄	1	2	yes	—
18	#16812679	C₄	1	1	yes	characteristic
19	#16812677	C₄	1	2	yes	—
20	#16812675	C₄	1	1	yes	characteristic
21	C₄×Dic₂₁₈	C₂²	1	2	yes	—
22	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
23	#16812656	C₂²	1	2	yes	—
24	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
25	C₄⋊#11724784	C₂²	1	1	yes	characteristic
26	C₄⋊#11724784	C₂²	1	2	yes	—
27	#16812628	C₂²	1	2	yes	—
28	#16812628	C₂²	1	2	yes	—
29	#16812628	C₂²	1	2	yes	—
30	#16812609	C₂²	1	2	yes	—
31	C₄×#11724784	C₂²	1	1	yes	characteristic
32	C₄×#11724784	C₂²	1	2	yes	—
33	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₀₉	C₂²	1	2	yes	—
36	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
39	#97×C₁₀₉	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
42	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
46	#13182012	—	2	4	no	—
47	C₂×#11724784	C₂³	1	1	yes	characteristic
48	C₂×#11724784	C₂³	1	2	yes	—
49	C₂×#11724784	C₂³	1	1	yes	characteristic
50	C₈×D₁₀₉	—	2	2	no	—
51	C₈×D₁₀₉	Q₈	1	4	yes	complemented
52	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
54	#13182008	C₂×C₄	1	2	yes	—
55	#13182008	C₂×C₄	1	2	yes	—
56	#13182008	C₂×C₄	1	2	yes	—
57	#13182008	C₂×C₄	1	2	yes	—
58	#13182008	C₂×C₄	1	2	yes	—
59	#13182006	C₂×C₄	1	2	yes	—
60	#13182006	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
64	C₄×Dic₁₀₉	C₂³	1	2	yes	—
65	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
68	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
70	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₄²×C₁₀₉	C₂³	1	2	yes	—
73	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₁₀₉	—	2	2	no	—
75	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₁₀₉	—	2	2	no	—
86	C₄×D₁₀₉	—	2	2	no	—
87	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
88	C₄×D₁₀₉	#41	1	2	yes	—
89	Dic₂₁₈	—	2	2	no	—
90	Dic₂₁₈	—	2	4	no	—
91	C₈×C₁₀₉	—	2	1	no	—
92	C₈×C₁₀₉	—	2	2	no	—
93	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
94	#11724784	—	2	1	no	—
95	#11724784	—	2	2	no	—
96	#11724784	C₂×Q₈	1	2	yes	—
97	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₀₉	—	2	1	no	—
99	C₂×D₁₀₉	—	2	1	no	—
100	C₂×D₁₀₉	C₄×Q₈	1	2	yes	—
101	C₄×C₁₀₉	—	2	2	no	—
102	C₄×C₁₀₉	—	2	1	no	—
103	C₄×C₁₀₉	—	2	2	no	—
104	C₄×C₁₀₉	#131	1	2	yes	—
105	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
107	Dic₁₀₉	—	2	2	no	—
108	Dic₁₀₉	—	2	2	no	—
109	Dic₁₀₉	—	2	1	no	—
110	Dic₁₀₉	C₄×Q₈	1	2	yes	—
111	Dic₁₀₉	#131	1	2	yes	—
112	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
113	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
114	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
115	D₁₀₉	—	2	2	no	—
116	#2960	—	109	1	no	maximal, Sylow
117	C₁₀₉	#2960	1	1	yes	characteristic, complemented, Sylow
118	#533	—	109	1	no	—
119	#446	—	109	2	no	—
120	#446	—	109	1	no	—
121	#446	D₁₀₉	1	1	yes	characteristic, complemented
122	C₈×Q₈	—	109	1	no	—
123	C₈×Q₈	—	109	2	no	—
124	C₈×Q₈	—	109	1	no	—
125	#424	—	109	2	no	—
126	#423	—	109	1	no	—
127	#405	—	109	2	no	—
128	C₂×C₄×C₈	—	109	1	no	—
129	C₂×M₁₆	—	109	2	no	—
130	C₂²×C₈	—	109	1	no	—
131	C₂²×C₈	—	109	1	no	—
132	C₄⋊Q₈	—	109	1	no	—
133	C₄⋊Q₈	—	109	1	no	—
134	#125	—	109	2	no	—
135	C₂²⋊Q₈	—	109	1	no	—
136	C₂²⋊Q₈	—	109	2	no	—
137	C₂²⋊Q₈	—	109	1	no	—
138	C₄×Q₈	—	109	2	no	—
139	C₄×Q₈	—	109	1	no	—
140	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
141	#117	—	109	2	no	—
142	C₂×C₄²	—	109	1	no	—
143	C₄⋊C₈	—	109	2	no	—
144	C₄⋊C₈	—	109	2	no	—
145	C₄⋊C₈	—	109	2	no	—
146	C₄⋊C₈	—	109	1	no	—
147	C₄⋊C₈	—	109	2	no	—
148	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
150	#97	—	109	2	no	—
151	#97	C₂×D₁₀₉	1	2	yes	—
152	C₄×C₈	—	109	1	no	—
153	C₄×C₈	—	109	1	no	—
154	C₄×C₈	—	109	2	no	—
155	C₄×C₈	—	109	1	no	—
156	C₄×C₈	—	109	2	no	—
157	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
158	C₂×Q₈	—	109	2	no	—
159	C₂×Q₈	—	109	1	no	—
160	C₂×Q₈	C₄×D₁₀₉	1	1	yes	characteristic
161	C₂²×C₄	—	109	1	no	—
162	C₂²×C₄	—	109	1	no	—
163	C₂²×C₄	—	109	1	no	—
164	M₁₆	—	218	4	no	—
165	C₂×C₈	—	109	1	no	—
166	C₂×C₈	—	109	4	no	—
167	C₂×C₈	—	109	1	no	—
168	C₂×C₈	—	109	2	no	—
169	C₂×C₈	—	218	2	no	—
170	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
172	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
173	C₄⋊C₄	—	109	2	no	—
174	C₄⋊C₄	—	109	2	no	—
175	C₄⋊C₄	—	109	2	no	—
176	C₄⋊C₄	—	109	2	no	—
177	C₄⋊C₄	—	109	2	no	—
178	C₄⋊C₄	—	109	2	no	—
179	C₄⋊C₄	—	109	1	no	—
180	C₄⋊C₄	C₄×D₁₀₉	1	2	yes	—
181	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
182	C₂²⋊C₄	—	109	2	no	—
183	C₂²⋊C₄	—	109	2	no	—
184	C₄²	—	109	1	no	—
185	C₄²	—	109	1	no	—
186	C₄²	—	109	1	no	—
187	C₄²	—	109	2	no	—
188	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
189	C₄²	C₂²×D₁₀₉	1	2	yes	—
190	C₂³	—	109	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	218	4	no	—
193	Q₈	—	218	2	no	—
194	C₂×C₄	—	109	1	no	—
195	C₂×C₄	—	109	1	no	—
196	C₂×C₄	—	109	2	no	—
197	C₂×C₄	—	109	2	no	—
198	C₂×C₄	—	109	1	no	—
199	C₂×C₄	—	109	2	no	—
200	C₂×C₄	—	109	2	no	—
201	C₂×C₄	—	218	2	no	—
202	C₂×C₄	—	218	2	no	—
203	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	2	no	—
210	C₈	—	109	2	no	—
211	C₈	—	218	1	no	—
212	C₈	—	218	2	no	—
213	C₈	Q₈×D₁₀₉	1	2	yes	—
214	C₂²	—	109	2	no	—
215	C₂²	—	218	1	no	—
216	C₂²	—	218	1	no	—
217	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	109	2	no	—
222	C₄	—	109	2	no	—
223	C₄	—	218	2	no	—
224	C₄	—	218	1	no	—
225	C₄	—	218	2	no	—
226	C₄	#16812707	1	2	yes	—
227	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
228	C₄	#41×D₁₀₉	1	1	yes	characteristic
229	C₂	—	218	2	no	—
230	C₂	C₄×Q₈×D₁₀₉	1	1	yes	characteristic
231	C₂	#131×D₁₀₉	1	1	yes	characteristic
232	C₂	C₂×#16812707	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#446×C₁₀₉	C₂
prime radical	C₂²×D₁₀₉	C₂²×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#446	D₁₀₉
hypocenter	C₁₀₉	#2960
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #20295156	C₂	semidirect
3. #20295154	C₂	semidirect
4. Q₈×#11724784	C₂	semidirect
5. #20295150	C₂	semidirect
6. #20295150	C₂	semidirect
11. #446×C₁₀₉	C₂	semidirect
51. C₈×D₁₀₉	Q₈	semidirect
117. C₁₀₉	#2960	semidirect
121. #446	D₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295580	2
#20295156	2
#20295154	2
Q₈×#11724784	2
#20295150	2
#20295051	2
#20295048	2
#20294983	2
C₄×C₈×D₁₀₉	2
#446×C₁₀₉	2
#2960	109